Question: Solve for $x$ : $ 8|x + 5| - 10 = -5|x + 5| + 7 $
Solution: Add $ {5|x + 5|} $ to both sides: $ \begin{eqnarray} 8|x + 5| - 10 &=& -5|x + 5| + 7 \\ \\ { + 5|x + 5|} && { + 5|x + 5|} \\ \\ 13|x + 5| - 10 &=& 7 \end{eqnarray} $ Add ${10}$ to both sides: $ \begin{eqnarray} 13|x + 5| - 10 &=& 7 \\ \\ { + 10} &=& { + 10} \\ \\ 13|x + 5| &=& 17 \end{eqnarray} $ Divide both sides by ${13}$ $ \dfrac{13|x + 5|} {{13}} = \dfrac{17} {{13}} $ Simplify: $ |x + 5| = \dfrac{17}{13}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{17}{13} $ or $ x + 5 = \dfrac{17}{13} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{17}{13} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{17}{13} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{17}{13} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $13$ $ x = - \dfrac{17}{13} {- \dfrac{65}{13}} $ $ x = -\dfrac{82}{13} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{17}{13} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{17}{13} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{17}{13} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $13$ $ x = \dfrac{17}{13} {- \dfrac{65}{13}} $ $ x = -\dfrac{48}{13} $ Thus, the correct answer is $x = -\dfrac{82}{13} $ or $x = -\dfrac{48}{13} $.